Invariant differential operators on H-type groups and discrete components in restrictions of complementary series of rank one semisimple groups - Danish National Research Database-Den Danske Forskningsdatabase

^{1} Department of Mathematics, Science and Technology, Aarhus University^{2} Chalmers University of Technology and Mathematical Sciences, Göteborg^{3} Department of Mathematics, Science and Technology, Aarhus University

DOI:

10.1007/s12220-014-9540-z

Abstract:

We explicitly construct a finite number of discrete components in the restriction of complementary series representations of rank one semisimple groups $G$ to rank one subgroups $G_1$. For this we use the realizations of complementary series representations of $G$ and $G_1$ on Sobolev spaces on the nilpotent radicals $N$ and $N_1$ of the minimal parabolics in $G$ and $G_1$, respectively. The groups $N$ and $N_1$ are of H-type and we construct explicitly invariant differential operators between $N$ and $N_1$. These operators induce the projections onto the discrete components. Our construction of the invariant differential operators is carried out uniformly in the framework of H-type groups and also works for those H-type groups which do not occur as nilpotent radical of a parabolic subgroup in a semisimple group.

Type:

Journal article

Language:

English

Published in:

Journal of Geometric Analysis, 2016, Vol 26, Issue 1, p. 118-142